The Development of Radon Transforms Associated to Compton Scatter Imaging Concepts

Abstract

For more than three decades, the exploitation of Compton scattering has led to significant progress in imaging the inner parts of an optically opaque object. In this imaging process, the Compton scattering angle plays an essential role as it labels the recorded data, suppressing the motion of the scanning camera, as well as generates new geometric manifolds on which radiation measurements are made and thereby introducing new relevant Radon transforms. The present paper is aimed to synthetize the conceptual development of Compton scattering imaging in order to gain a constructive overview and possibly induce a further innovative growth to it. We believe this may lay foundation to new mathematical tools as well as lead to a great diversity of applications.

An inversion of the conical Radon transform arising in the Compton camera with helical movement

“…f ( ) will be reconstructed from f ( , k (u)) , when (u) = (cos u, sin u, u) and the numerical phantom density function f p is given by the characteristic function of the following geometry: which is the special case given in (20).…”

“…4b, three-dimensional relative RMS errors are compared for a filtered backprojection error from data f and the added Hilbert transform error from data f . The data f and the Radon transform f for the density function in (20) are given by (21) and (22), respectively. Therefore, the error of the Hilbert transform is approximated by the difference of these two errors, which is considered to be quite small as compared to the filtered backprojection error alone.…”

“…Ever since the Compton camera, also known as an electronically collimated -camera, was first introduced for use in Single Photon Emission Computed Tomography (SPECT), a Radon-type transform that assigns a function to its surface integral over various sets of cones has attracted substantial attention [2,3,5,8,9,12,13,16,17,20,23,27,[29][30][31]. For the generalized Radon transform including toric Radon transform, see [22].…”

The conical Radon transform with vertices on triple line segments

“…Also, as a special two-dimensional version of the conical Radon transform, the V-line transform is studied in [3,14]. For recent reviews of conical Radon transforms see [2,22,28]. In this paper we restrict the cones of integration to a three-dimensional submanifold of conical surfaces associated to linear detectors where the vertices and the axes directions are restricted to one dimension.…”

Orthogonal function series formulae for inversion of the conical Radon transform with a fixed central axis

“…Thus, the inversion provides the reconstructed image in single scattering optical tomography (for further details, see [1,9]). Many previous studies [11, 21-23, 28, 34, 36] have investigated various types of conical Radon transforms, which have also been reviewed comprehensively [26,33]. Most of these previous studies considered the conical Radon transform with a variable central axis for the cone of integration.…”